{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/116162"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/116162","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Local symplectic groupoids and the SGA equation","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","abstract_has_math":false,"creators":["Zhang, Yuxuan"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Fernandes, Rui Loja","Kerman, Ely","Lerman, Eugene","Pascaleff, James"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-08","date_published":"2022-08","updated_at":"2026-07-22T22:24:55Z","subjects":["local symplectic groupoids","integrability of Poisson manifolds","SGA equation"],"languages":["en","eng"],"rights":["Copyright 2022 Yuxuan Zhang"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/116162","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Fernandes, Rui Loja","Kerman, Ely","Lerman, Eugene","Pascaleff, James"]},{"key":"dc:creator","label":"Author","values":["Zhang, Yuxuan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-08","2022-06-27"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["local symplectic groupoids","integrability of Poisson manifolds","SGA equation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2022 Yuxuan Zhang"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/116162"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","The student, Yuxuan Zhang, accepted the attached license on 2022-06-15 at 10:09.","The student, Yuxuan Zhang, submitted this Dissertation for approval on 2022-06-15 at 10:17.","This Dissertation was approved for publication on 2022-06-27 at 09:49.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18077 on 2022-11-15 at 17:37:39","This thesis discusses several problems related to local symplectic groupoids. In Chapter 1, we prove that if a local symplectic groupoid has uniformly discrete associators, then its associative completion is a symplectic groupoid. It follows that a Poisson manifold is integrable if and only if any of its local integrations has uniformly discrete associators. In Chapter 2, we construct a local symplectic groupoid integrating the Heisenberg-Poisson manifold which is not 6-associative. In Chapter 3, we give the conditions for a function to be the generating function for some local symplectic groupoid structure on the cotangent bundle, both for a coordinate space and for an abstract manifold. We also compare different notions of generating functions and analyze the role of the SGA equation. In Chapter 4, we show that the algebraic equation in the SGA equation is equivalent to a groupoid 2-cocycle condition. Under mild assumptions on the local symplectic groupoid, we find a groupoid 2-cocycle which under the van Est map yields the underlying Poisson bivector."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Local symplectic groupoids and the SGA equation"]}]}],"canonical_facts":{"dc:contributor":["Fernandes, Rui Loja","Kerman, Ely","Lerman, Eugene","Pascaleff, James"],"dc:creator":["Zhang, Yuxuan"],"dc:date":["2022-08","2022-06-27"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","The student, Yuxuan Zhang, accepted the attached license on 2022-06-15 at 10:09.","The student, Yuxuan Zhang, submitted this Dissertation for approval on 2022-06-15 at 10:17.","This Dissertation was approved for publication on 2022-06-27 at 09:49.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18077 on 2022-11-15 at 17:37:39","This thesis discusses several problems related to local symplectic groupoids. In Chapter 1, we prove that if a local symplectic groupoid has uniformly discrete associators, then its associative completion is a symplectic groupoid. It follows that a Poisson manifold is integrable if and only if any of its local integrations has uniformly discrete associators. In Chapter 2, we construct a local symplectic groupoid integrating the Heisenberg-Poisson manifold which is not 6-associative. In Chapter 3, we give the conditions for a function to be the generating function for some local symplectic groupoid structure on the cotangent bundle, both for a coordinate space and for an abstract manifold. We also compare different notions of generating functions and analyze the role of the SGA equation. In Chapter 4, we show that the algebraic equation in the SGA equation is equivalent to a groupoid 2-cocycle condition. Under mild assumptions on the local symplectic groupoid, we find a groupoid 2-cocycle which under the van Est map yields the underlying Poisson bivector."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/116162"],"dc:language":["en","eng"],"dc:rights":["Copyright 2022 Yuxuan Zhang"],"dc:subject":["local symplectic groupoids","integrability of Poisson manifolds","SGA equation"],"dc:title":["Local symplectic groupoids and the SGA equation"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:55Z"}